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<h1 class="ltx_title ltx_title_section">PSF model: Integrated form of a symmetric 2D Gaussian function</h1>

<div id="Sx1.p1" class="ltx_para">
<p class="ltx_p">The integrated form of a <a href="SymmetricGaussianEstimatorUI.html" title="" class="ltx_ref">symmetric two-dimensional Gaussian function</a>
can be used to help to take into account the discrete nature of pixels
present in digital cameras <cite class="ltx_cite">[<a href="#bib.bib31" title="Fast, single-molecule localization that achieves theoretically minimum uncertainty" class="ltx_ref">2</a>, <a href="#bib.bib15" title="Simultaneous multiple-emitter fitting for single molecule super-resolution imaging" class="ltx_ref">1</a>]</cite>. Assuming a
uniform distribution of pixels with unit size, a single molecule intensity
profile can be expressed as</p>
</div>
<div id="Sx1.p2" class="ltx_para">
<table id="Sx1.Ex1" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.Ex1.m1" class="ltx_Math" style="vertical-align:-7px" src="mi/mi112.png" width="256" height="22" alt="\mathrm{PSF_{IG}}\left(x,y\mid\boldsymbol{\theta}\right)=\theta_{N}E_{x}E_{y}+%
\theta_{b}\,,"></td>
<td class="ltx_eqn_pad"></td>
</tr>
</table>
<p class="ltx_p">where <img id="Sx1.p2.m1" class="ltx_Math" style="vertical-align:-6px" src="mi/mi114.png" width="123" height="21" alt="\mathrm{PSF_{IG}}\left(x,y\mid\boldsymbol{\theta}\right)">
gives the expected photon count at the integer pixel position <img id="Sx1.p2.m2" class="ltx_Math" style="vertical-align:-6px" src="mi/mi60.png" width="45" height="21" alt="\left(x,y\right)">
for a vector of parameters <img id="Sx1.p2.m3" class="ltx_Math" style="vertical-align:-7px" src="mi/mi113.png" width="167" height="22" alt="\boldsymbol{\theta}=\left[\theta_{x},\theta_{y},\theta_{\sigma},\theta_{N},%
\theta_{b}\right]">
and</p>
</div>
<div id="Sx1.p3" class="ltx_para">
<table id="Sx1.EGx1" class="ltx_equationgroup ltx_eqn_eqnarray">

<tr id="Sx1.Ex2" class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_td ltx_align_right"><img id="Sx1.Ex2.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi107.png" width="26" height="18" alt="\displaystyle E_{x}"></td>
<td class="ltx_td ltx_align_center"><img id="Sx1.Ex2.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi1.png" width="18" height="11" alt="\displaystyle="></td>
<td class="ltx_td ltx_align_left"><img id="Sx1.Ex2.m3" class="ltx_Math" style="vertical-align:-19px" src="mi/mi110.png" width="340" height="48" alt="\displaystyle\frac{1}{2}\erf\left(\frac{x-\theta_{x}+\frac{1}{2}}{\sqrt{2}%
\theta_{\sigma}}\right)-\frac{1}{2}\erf\left(\frac{x-\theta_{x}-\frac{1}{2}}{%
\sqrt{2}\theta_{\sigma}}\right)\,,"></td>
<td class="ltx_eqn_pad"></td>
</tr>
<tr id="Sx1.Ex3" class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_td ltx_align_right"><img id="Sx1.Ex3.m1" class="ltx_Math" style="vertical-align:-7px" src="mi/mi109.png" width="26" height="21" alt="\displaystyle E_{y}"></td>
<td class="ltx_td ltx_align_center"><img id="Sx1.Ex3.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi1.png" width="18" height="11" alt="\displaystyle="></td>
<td class="ltx_td ltx_align_left"><img id="Sx1.Ex3.m3" class="ltx_Math" style="vertical-align:-19px" src="mi/mi111.png" width="337" height="48" alt="\displaystyle\frac{1}{2}\erf\left(\frac{y-\theta_{y}+\frac{1}{2}}{\sqrt{2}%
\theta_{\sigma}}\right)-\frac{1}{2}\erf\left(\frac{y-\theta_{y}-\frac{1}{2}}{%
\sqrt{2}\theta_{\sigma}}\right)\,."></td>
<td class="ltx_eqn_pad"></td>
</tr>
</table>
<p class="ltx_p">The entries of the vector <img id="Sx1.p3.m1" class="ltx_Math" style="vertical-align:-2px" src="mi/mi56.png" width="15" height="16" alt="\boldsymbol{\theta}"> are as follows: <img id="Sx1.p3.m2" class="ltx_Math" style="vertical-align:-5px" src="mi/mi67.png" width="21" height="19" alt="\theta_{x}">
and <img id="Sx1.p3.m3" class="ltx_Math" style="vertical-align:-7px" src="mi/mi68.png" width="21" height="21" alt="\theta_{y}"> are the sub-pixel molecular coordinates, <img id="Sx1.p3.m4" class="ltx_Math" style="vertical-align:-5px" src="mi/mi92.png" width="22" height="19" alt="\theta_{\sigma}">
is the imaged size of the molecule, <img id="Sx1.p3.m5" class="ltx_Math" style="vertical-align:-5px" src="mi/mi63.png" width="26" height="19" alt="\theta_{N}"> corresponds to the
total number of photons emitted by the molecule, and <img id="Sx1.p3.m6" class="ltx_Math" style="vertical-align:-5px" src="mi/mi66.png" width="20" height="19" alt="\theta_{b}">
corresponds to the background signal level.</p>
</div>
<div id="Sx1.SSx1" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">See also</h2>

<div id="Sx1.SSx1.p1" class="ltx_para">
<ul id="I1" class="ltx_itemize">
<li id="I1.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i1.p1" class="ltx_para">
<p class="ltx_p"><a href="PSF.html" title="" class="ltx_ref">Point-spread function (PSF)</a></p>
</div>
</li>
<li id="I1.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i2.p1" class="ltx_para">
<p class="ltx_p"><a href="Fitting.html" title="" class="ltx_ref">Fitting point-spread function models</a></p>
</div>
</li>
<li id="I1.i3" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i3.p1" class="ltx_para">
<p class="ltx_p"><a href="FittingRegion.html" title="" class="ltx_ref">Definition of the fitting region</a></p>
</div>
</li>
<li id="I1.i4" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i4.p1" class="ltx_para">
<p class="ltx_p"><a href="LocalizationUncertainty.html" title="" class="ltx_ref">Localization uncertainty</a></p>
</div>
</li>
<li id="I1.i5" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i5.p1" class="ltx_para">
<p class="ltx_p"><a href="CrowdedField.html" title="" class="ltx_ref">Multiple-emitter fitting analysis</a></p>
</div>
</li>
</ul>
</div>
</div>
</div>
<div id="bib" class="ltx_bibliography">
<h1 class="ltx_title ltx_title_bibliography">References</h1>

<ul id="L1" class="ltx_biblist">
<li id="bib.bib15" class="ltx_bibitem ltx_bib_article">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[1]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">F. Huang, S. L. Schwartz, J. M. Byars and K. A. Lidke</span><span class="ltx_text ltx_bib_year">(2011)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">Simultaneous multiple-emitter fitting for single molecule super-resolution imaging</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_journal">Biomedical Optics Express</span> <span class="ltx_text ltx_bib_volume">2</span> (<span class="ltx_text ltx_bib_number">5</span>), <span class="ltx_text ltx_bib_pages"> pp. 1377–93</span>.
</span>
<span class="ltx_bibblock">External Links: <span class="ltx_text ltx_bib_links"><a href="http://dx.doi.org/10.1364/BOE.2.001377" title="" class="ltx_ref doi ltx_bib_external">Document</a></span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.p1" title="PSF model: Integrated form of a symmetric 2D Gaussian function" class="ltx_ref"><span class="ltx_text ltx_ref_title">PSF model: Integrated form of a symmetric 2D Gaussian function</span></a>.
</span>
</li>
<li id="bib.bib31" class="ltx_bibitem ltx_bib_article">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[2]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">C. S. Smith, N. Joseph, B. Rieger and K. A. Lidke</span><span class="ltx_text ltx_bib_year">(2010)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">Fast, single-molecule localization that achieves theoretically minimum uncertainty</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_journal">Nature Methods</span> <span class="ltx_text ltx_bib_volume">7</span> (<span class="ltx_text ltx_bib_number">5</span>), <span class="ltx_text ltx_bib_pages"> pp. 373–5</span>.
</span>
<span class="ltx_bibblock">External Links: <span class="ltx_text ltx_bib_links"><a href="http://dx.doi.org/10.1038/nmeth.1449" title="" class="ltx_ref doi ltx_bib_external">Document</a>,
<span class="ltx_text issn ltx_bib_external">ISSN 1548-7105</span></span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.p1" title="PSF model: Integrated form of a symmetric 2D Gaussian function" class="ltx_ref"><span class="ltx_text ltx_ref_title">PSF model: Integrated form of a symmetric 2D Gaussian function</span></a>.
</span>
</li>
</ul>
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